Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the trigonometric identity \(\sin 2 x=2 \sin x \cos x\) along with the Product Rule to find \(D_{x} \sin 2 x\).

Step-by-Step Solution

Answer

The derivative of \( \sin 2x \) is \( 2 \cos 2x \).

Step 1: Recall the Trigonometric Identity

We are given the trigonometric identity \( \sin 2x = 2 \sin x \cos x \). This will be our starting point to find the derivative.

Step 2: Differentiate Using the Product Rule

The expression \( \sin 2x = 2 \sin x \cos x \) is a product of two functions: \( \sin x \) and \( \cos x \). The Product Rule states that \( D_x (uv) = u'v + uv' \) for functions \( u(x) \) and \( v(x) \). Here, let \( u = \sin x \) and \( v = \cos x \).

Step 3: Compute Individual Derivatives

Differentiate \( u = \sin x \) to get \( u' = \cos x \) and \( v = \cos x \) to get \( v' = -\sin x \).

Step 4: Apply the Product Rule

Using the Product Rule, the derivative is \( D_x (2 \sin x \cos x) = 2 (u'v + uv') = 2( \cos x \cdot \cos x + \sin x \cdot (-\sin x) ) = 2(\cos^2 x - \sin^2 x) \).

Step 5: Simplify the Expression

Simplifying the expression, we have \( 2 \cos^2 x - 2 \sin^2 x = 2 (\cos^2 x - \sin^2 x) \), which is a well-known trigonometric identity for \( \cos 2x \).

Step 6: Write the Final Derivative

The derivative of \( \sin 2x \) is \( 2 \cos 2x \) as \( D_x \sin 2x = 2 \cos 2x \).

Key Concepts

Product Rule

The Product Rule is an essential technique in calculus used for finding derivatives of functions that are the product of two simpler functions. When you have two functions, let's call them \( u(x) \) and \( v(x) \), and you want to find the derivative of their product \( u(x)v(x) \), you need to use the Product Rule.

The Product Rule can be stated mathematically as:

• \( D_x(uv) = u'v + uv' \)

This means to find the derivative of their product, you first differentiate \( u \) while keeping \( v \) unchanged, then add the result to the product of \( u \) unchanged and the derivative of \( v \).
Here is a step-by-step approach:

• Find \( u' = \frac{du}{dx} \)
• Keep \( v \) the same
• Multiply \( u' \) by \( v \)
• Find \( v' = \frac{dv}{dx} \)
• Keep \( u \) the same
• Multiply \( u \) by \( v' \)
• Add the two products together

This rule is especially helpful when working with trigonometric functions, like \( 2\sin x \cos x \), as it simplifies the differentiation process.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equality are defined. These identities are essential tools in calculus differentiation, providing shortcuts and simplifications. For example, the double angle identity \( \sin 2x = 2 \sin x \cos x \) is crucial for transforming complex trigonometric expressions into simpler, more manageable forms. By expressing one trigonometric function in terms of others, you streamline the process of differentiation and integration.Utilizing identities like \( \sin^2 x + \cos^2 x = 1 \) or transforming expressions with \( \cos 2x = \cos^2 x - \sin^2 x \), can make finding derivatives and integrals more straightforward. Knowing these identities empowers you to manipulate and simplify expressions, making the calculus operations manageable by significantly reducing the effort required.

Calculus Differentiation

Calculus differentiation is a mathematical process that calculates how a function changes at any point or the function's rate of change. It is one of the core concepts allowing us to analyze how quantities vary together.For instance, when differentiating trigonometric functions like \( \sin 2x \), it involves using various rules and identities:

• The Chain Rule, applicable when dealing with function compositions
• The Product Rule, for multiplying functions
• Trigonometric Identities, to simplify and manipulate functions

The goal is to determine the derivative function, which tells you how the original function behaves. For example, by finding that \( D_x(\sin 2x) = 2\cos 2x \), you understand the rate at which \( \sin 2x \) changes with respect to \( x \).
Differentiation underpins many real-world applications, such as calculating speed from position, among others, by giving insight into patterns of change.